The integral Bessel functions of the zero order are inverse transforms of
the following Laplace transforms [1]}
\(\begin{array}{l}\displaystyle \mathcal {L}^{-1}\left[ \frac{\sinh ^{-1}s}{s}\right] =\mathrm {Ji}_{0}\left(t\right) , \\\displaystyle \mathcal {L}^{-1}\left[ \frac{\left( \sinh ^{-1}s\right) ^{2}}{s}\right] =\mathrm {Yi}_{0}\left( t\right) , \\\displaystyle \mathcal {L}^{-1}\left[ \mathrm {\ln }\left( s+\sqrt{s^{2}+1}\right) -\frac{\pi i}{2}\right] =\mathrm {Ii}_{0}\left( t\right) , \\\displaystyle \mathcal {L}^{-1}\left[ \frac{\left( \cosh ^{-1}s\right) ^{2}}{2s}+\frac{\pi ^{2}}{8s}\right] =\mathrm {Ki}_{0}\left( t\right) .\end{array}\) 
